To solve for the complex number $z$ that satisfies the equation

$z(1 + \sqrt{2} i) = 1 - \sqrt{2} i$

we'll go through the following steps:

1. Rewrite the equation to isolate $z$: $z = \frac{1 - \sqrt{2} i}{1 + \sqrt{2} i}$

2. Multiply the numerator and denominator by the complex conjugate of the denominator to simplify: $z = \frac{(1 - \sqrt{2} i)(1 - \sqrt{2} i)}{(1 + \sqrt{2} i)(1 - \sqrt{2} i)}$

3. Simplify the denominator: $(1 + \sqrt{2} i)(1 - \sqrt{2} i) = 1^2 - (\sqrt{2} i)^2 = 1 - (-2) = 1 + 2 = 3$

   So, the denominator simplifies to 3.

4. Expand the numerator: $(1 - \sqrt{2} i)(1 - \sqrt{2} i) = 1 - 2\sqrt{2} i + (\sqrt{2} i)^2 = 1 - 2\sqrt{2} i - 2 = -1 - 2\sqrt{2} i$

5. Combine everything: $z = \frac{-1 - 2\sqrt{2} i}{3} = -\frac{1}{3} - \frac{2\sqrt{2}}{3} i$

Thus, the solution is:

$z = -\frac{1}{3} - \frac{2\sqrt{2}}{3} i$

Would you like further details on any step or have any other questions?

Here are some related questions:

1. How can you use complex conjugates to simplify division in complex numbers?
2. What are the steps to find the modulus of a complex number like $1 + \sqrt{2}i$?
3. How can polar form be helpful in solving similar complex equations?
4. What is the geometric interpretation of complex multiplication?
5. How do complex conjugates help in rationalizing denominators with complex numbers?

Tip: When dividing complex numbers, multiplying by the complex conjugate of the denominator is a reliable way to eliminate imaginary components from the denominator.